Multiple Regression Analysis

The focus is on understanding the individual contributions of independent variables (predictors) in explaining the variance in the dependent variable.
Tolerance is examined as a statistic to identify multicollinearity issues among independent variables.
Other statistics, including those for unique variance, are considered.

Course quality is the dependent variable (Y), measured on a Likert scale from 1 to 5.
Independent variables (Xs) include:
- Enrolment number
- Exam quality
- Grade expected
- Lecturer's knowledge
- Lecturer's ability
The goal is to determine the combination of Xs that best explains the variance in Y (R-squared) and to identify the strongest individual predictors.
The interrelationship between the X variables is considered, as overlap may influence their predictive power on Y.
Simple correlations(R values) alone are potentially misleading due to the absence of other variables in the model.

Green numbers with asterisks indicate statistically significant predictors of course quality.
Single asterisk (*) indicates significance at the p < 0.05 level.
Double asterisk (**) indicates significance at the p < 0.01 level or lower.

R values represent simple correlations between each independent variable and course quality.
R-squared values indicate the proportion of variance in the dependent variable explained by each independent variable.
To calculate the proportion of variance explained, square the R value (e.g., $R = 0.6$ , $R^2 = 0.36$ , explaining 36% of the variation).

Significant correlations exist between:
*Exam quality and Enrolment number
*Grade expected and Exam quality
*Lecturer ability and Exam quality
Multicollinearity is a concern in multiple regression if independent variables are highly correlated.

Adding up individual R-squared values can result in a total greater than 100% due to overlap and redundant contributions among the predictors.
Example: Sum of individual R-squared values is 1.613 (161.3%), while the regression model explains 76% of the variance in course quality.
The adjusted R-squared accounts for sample size, especially when the sample size is small (e.g., 50 in this case).

Statistical packages can run each independent variable(X) as the dependent variable and the other variables as independent variables to predict it.
Example: Using four independent variables to predict enrolment explains 35% of its variance.
Tolerance is calculated as $1 - R^2$ .
Acceptable tolerance values should be greater than 0.1.
Tolerance indicates how much variance in an independent variable is not explained by the other variables.
High tolerance values (e.g., greater than 0.85) indicate that the constructs are very different.

Regression tables include predictors, unstandardized beta coefficients, standardized beta coefficients, R value (zero-order correlation), unique correlation (SR), partial correlation, tolerance, and T value.
Tolerance indicates the potential contribution of a variable, considering how it relates to other independent variables.
A low tolerance (e.g., 0.12) means the variable does not share much with the other IVS while the unique percentage, explains a big chunk of the dependent variable.

Semi partial correlation (SR) indicates the unique contribution of each predictor.
$SR^2$ is more informative than SR because it indicates the proportion of variance uniquely explained by the variable.

Formula: $SR^2 = \text{Unique explanation by } X<em>1$ , partially out the unique from $X</em>2$ and the shared value with $X_2$ .
This formula isolates the unique contribution of $X_1$ .

Exam quality initially has an R of 0.6 ( $R^2 = 36\%$ .)
Unique correlation drops to 0.06 ( $SR^2 = 0.0036$ , less than 1%) after including other variables.
Lecturer teaching ability initially explains 64%, which decreases to 18% or 19% uniquely.
Total R-squared for the model is 75.5% or 76%.
The two most significant predictors (lecturer knowledge and lecturer ability) explain the highest unique variance (0.07 and 0.18, respectively).

Include the number of independent variables, the outcome variable, R-squared, F value, and p-value.
Report SR values to indicate the unique contribution of significant variables.

Compare zero-order correlations with semi partial correlations to assess the impact of including other independent variables.
Tolerance of 0.10 is the threshold for identifying multicollinearity issues.
Semi partial correlation helps understand the unique contribution of each independent variable.

A basic guideline is to start with 50 participants and add five participants for each predictor.
For five predictors, a sample size of $50 + (5 \times 5) = 75$ is needed.
G Power three is a more comprehensive tool for calculating power and sample size.